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2. Consider the equation 11.y+ay+r=0. (a) Show that the following statement is FALSE: "For all a,TER, there is a unique YeR such that r.y+ay+r=0." (b) Find a set of real numbers a such that the following statement is TRUE: "For all reR, there is a unique yer such that r.y+ay+r=0." Justify your answer.

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Jean Keeling
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2 Feb 2018
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8. Determine if each of the following statements is true or false. Clearly indicate your answer for each part by stating if it is true or false no justification is required). Each part is worth 2 marks for a correct answer; -1 for an incorrect answer to discourage guessing) (a) If A and B are n x n matrices, then rank(BA) rank(A). (b) If {A, B, C} is a linearly independent set of n x m matrices, then {AT, BT,CT} is a linearly independent set of m x n matrices. (C) W = {A E Rnxn A is invertible } is a subspace of Rnxn, where Rnxn denotes the vector space of real n x n matrices. (d) If xi and X2 are orthogonal vectors in R3 then it is always possible to find a unique vector X3 such that {x1, X2, X3} is an orthogonal basis for R3. (e) Let W be a subspace of R" and let x ER". Then projw(x - projw(x)) = 0.

[1] 1. (a) Solve 2 – 31 < 7. Solution: We have – 7<-3 < 7 which gives -4 0 on (-0, 1] U [2,). Next, we need to see where Vr2 – 3r +2 -1 > 0. If r2 - 3.c + 2 – 1 > 0, then V.22 – 3.0 +2 > . Observe that this is definitely true if r < 0. If r > 0, then squar- ing both sides we get 22 - 3.+2 > 1 2 > 3. V A VICO Thus, the domain of f is (-00, ). [2] (c) Find the exact value of tan(sec-14). Solution: Let y = sec-14. Then, sec y = 4. From the diagram we get that tan(sec-? 4) = tan y = 15 [2] (d) Find all r such that sin(2.x) = COS I. Solution: We have 2 sin r cos r = COS I, SO 0 = 2 sin Cos I -Cos I = cos .r (2 sin c - 1). Hence, sin(2x) = cost when cos x = 0, or when sin r = Thus, r= +ik, kez, I= +2nk, ke Z, or r= +2nk, kez [2] (e) State the precise mathematical definition of lim f(x) = 0. Solution: For every e > 0 there exists a 8 >0 such that if 0 < x-al <, then f(c) >

4. Find all possible values of the constants a and b such that f(2) is continuous for all real values of r, if f(c) is defined by: ac? if < -2 f(x) = {2|- 11 if -2

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